It seems to me that there is an issue with the epistemology of domains of quantification that has important implications for the epistemology and semantics of math generally, and which has received less attention than it deserves. In quick outline, the point is this:
A quantificational sentence has a determinate meaning only if there is some determinate fact of the matter as to what its domain of quantification is.So one knows what one is saying with such a sentence only if one knows what domain one is quantifying over. If we are discussing anything as complex as the reals - equivalently second order arithmetic - and mean to quantify over the "intended model" - that is, do not specify some constructable model as our domain - then we do not know what we are quantifying over.
Thus, we do not know what we are saying when we make claims with second order arithmetic quantifiers.
I take the first two steps to be pretty uncontroversial. We all know that a claim like "there is no more beer" can, in context, be perfectly correct. Because one is not quantifying over all beer in the universe, but merely the beer in the house. But suppose Jones is in a house which has a rather inaccessible cellar. There is no beer in the upstairs fridge but there is in the cellar. And in context, there is no fact of the matter whether his utterance "there is no more beer" is quantifying over the fridge or the whole house. I take it that it just follows that there is no fact of the matter whether the sentence is true, and thus no fact of the matter as to what it means. And if you utter a sentence not knowing which domain you are quantifying over, or the extent of the domain you mean, then you don't know what you said.
This is not just a special case of normal worries about the epistemology of mathematics. I can't offer anything like a complete account here, but I am something of a structuralist, and what I've called elsewhere "broadly inferentialist". The latter means that one understands a sentence insofar as one understands the various ways it is properly used, in particular the difference its various uses make to other linguistic proprieties. Structuralism implies that the difference a given mathematical assertion makes structurally - the difference its truth makes to the propriety of other assumptions about the relevant mathematical structure - are all the differences that matter. Once we understand what structure we are dealing with, in the sense of understanding all the differences implied by engaging with this structure rather than another, we understand all there is to the claims of mathematics. Finally, I believe in the priority of the sentential - more generally the priority of the whole speech act - over the subsentential. So to understand a term is to understand the sentences it can occur in. So once I grasp - whatever exactly that means - the relevant structure of the integers, I know what 7 is in the only sense of "knowing what 7 is" that makes sense.
I think this is pretty much how mathematicians think about things, but obviously I'm not arguing for that here. But if you buy anything like that, then we have a fine grasp of, for example, the integers, and likely a lot more. Indeed, while there are lots of inteeresting questions about what counts as a construction, I take it that when one has constructed a structure, one has demonstrated the relevant grasp of what it is. Hence, to quantify over a constructed domain is unproblematic.
But there's the issue. We can't construct "all the reals".
We have all sorts of constuctable models, some of them very nice and intuitive, some completely bizarre. For any model, constructable or not, we know that there are lots of other "non-standard" models that are quite different from it. So how do we know what we are quantifying over when we make a claim about all reals? We can say "I mean ALLLLLL reals." Swell, but how does that sortal get a determinate meaning. No use theory is going to help because the point about non-constructability is that there is no concrete use that settles the matter. And obviously causal aspects of meaning don't apply here.
Note this is also not a matter of some ontological scruple in favor of constructable models. I'll give you any level of platonism you want. The point is not that there aren't enough mathematical things out there, but that if there are any, there are too many. If there is one nonconstructable model of second order arithmetic, there are tons of them. Again, the issue is which one we are quantifying over since there seems to be no coherent sense in which our language is related to one more than another.
One further complication: For *some* sentences of second order arithmetic, we can safely assert them without knowing what the domain is. This is because they are true in all models. So we may not know precisely what the sentence is saying, that is what it is quantifying over, but in a supervaluational sense we can know that it is true. Whatever model we are talking about, this sentence is true. But not all sentences are like this. Schoenfield proved that all well-founded models of 2OA agree on the truth value of all Sigma 1-2 sentences (that is sentences limited in their complexity to two alternating quantifiers over reals), but there is no such fact for more complex sentences.
So our ignorance of our domain has implications for which sentences are true. And if a sentence is true under one interpretation and false under another, it has different meanings under them. And if we don't know which of these interpretations we intend, then we don't know what we mean.
I am inclined to think that this is a really serious issue. And that in the end it requires us to come up with a new conception of meaning for, at least, mathematical language - one that is less tied to fixed languages, models, and axiomatic theories. But that's a much bigger issue that I hope one day to have something written on.
(These thoughts, and all future writing on the subject, comes from joint work over the last 7 years with a mathematician friend K. Joseph Mourad.)